Exercise · ZW-LAGRANGE-MULTIPLIERS-010 · ★★★ Advanced
A solution of the Lagrange system has \(\lambda=0\). What does that imply?
Choose one option.
Show answer
- A. \(\nabla f=\mathbf 0\) there: it is also a critical point of \(f\)
Explanation
If \(\lambda=0\), the equation \(\nabla f=\lambda\nabla g\) gives \(\nabla f=\mathbf 0\): the constraint does not affect \(f\) and the point is already critical for unconstrained \(f\). We cannot tell whether it is a minimum.
Review the concept: Lagrange multipliers
The method of Lagrange multipliers finds the extrema of a function \(f\) on the curve \(g=k\) among the points where the gradient of \(f\) is a multiple \(\lambda\) of the gradient of \(g\).
\[\nabla f=\lambda\,\nabla g,\qquad g=k\]
More practice problems: lagrange multipliers
- ZW-LAGRANGE-MULTIPLIERS-001 Find the maximum value of f(x,y)=x+y on the circle x^2+y^2=2.
- ZW-LAGRANGE-MULTIPLIERS-002 Find the minimum value of f(x,y)=x^2+y^2 on the line x+2y=5.
- ZW-LAGRANGE-MULTIPLIERS-003 To maximize f(x,y)=xy subject to x+y=6, which system does the method set up?
- ZW-LAGRANGE-MULTIPLIERS-004 Find the maximum value of f(x,y)=xy on the line 2x+y=12.
- ZW-LAGRANGE-MULTIPLIERS-005 When minimizing f(x,y)=x^2+y^2 subject to x+y=4, what is λ at the optimal point?
- ZW-LAGRANGE-MULTIPLIERS-006 Find the maximum value of f(x,y)=xy on the circle x^2+y^2=8.
- ZW-LAGRANGE-MULTIPLIERS-007 Find the minimum value of f(x,y)=x+2y on the circle x^2+y^2=5.
- ZW-LAGRANGE-MULTIPLIERS-008 Find the minimum value of f(x,y,z)=x^2+y^2+z^2 on the plane x+y+z=3.